Counting multicurves on combinatorial surfaces and integrals of volumes of polytopes

Combinatorial surfaces can be seen as the limit of bordered hyperbolic surfaces with large boundaries, and we can study the problem of counting multicurves on them.

For a given combinatorial structure, multicurves correspond to integral points in a polytope defined in terms of ribbon graphs, and the asymptotic number of multicurves of very large length is controlled by the volume of those polytopes. This volume is a function on the moduli space of bordered surfaces, and motivated by questions from random geometry, one can ask for which s is this function $L^s$. The analog of this question can be posed in the hyperbolic case, and the answer is: $s\le 2$. In the hyperbolic case, recent results of Arana-Herrera show $L^2$. In the combinatorial case, we show using methods from convex geometry and analysis of divergences from subgraphs that the answer is $s<s_{g,n}$ for an explicit $s_{g,n}$ depending on the genus $g$ and the number of boundaries $n$. This leads to various (analytic and arithmetic) questions around such enumeration problems, which can be compared to questions about Feynman integrals.

This is a joint work with Charbonnier, Delecroix, Giacchetto and Wheeler.